Vector Algebra
Scalar products and vector components
Grade 12
Question:
<p>The 3-dimensional vectors <strong>v</strong><sub>1</sub>, <strong>v</strong><sub>2</sub>, <strong>v</strong><sub>3</sub> satisfying \(\mathbf{v}_1 \cdot \mathbf{v}_1 = 4\), \(\mathbf{v}_1 \cdot \mathbf{v}_2 = -2\), \(\mathbf{v}_1 \cdot \mathbf{v}_3 = 6\), \(\mathbf{v}_2 \cdot \mathbf{v}_2 = 2\), \(\mathbf{v}_2 \cdot \mathbf{v}_3 = -5\), \(\mathbf{v}_3 \cdot \mathbf{v}_3 = 29\), then <strong>v</strong><sub>3</sub> may be</p>
<p>(a) \(-3\mathbf{i} + 2\mathbf{j} \pm 4\mathbf{k}\)</p>
<p>(b) \(3\mathbf{i} - 2\mathbf{j} \pm 4\mathbf{k}\)</p>
<p>(c) \(-2\mathbf{i} + 3\mathbf{j} \pm 4\mathbf{k}\)</p>
<p>(d) \(2\mathbf{i} + 3\mathbf{j} \pm 4\mathbf{k}\)</p>
Step-by-Step Solution
Key Concept: Express v₃ as a linear combination of v₁ and v₂ using the given dot product constraints, then verify which option satisfies all six conditions simultaneously.
Step 1: Set up the linear combination. Since we have constraints on dot products with v_1 and v_2, assume v_3 = av_1 + bv_2 + w, where w is orthogonal to both v_1 and v_2. Step 2: Use dot product constraints with v_1. v_3·v_1 = a(v_1·v_1) + b(v_2·v_1) + w·v_1 = 4a - 2b = 6. This gives: 4a - 2b = 6, or 2a - b = 3. (Equation 1) Step 3: Use dot product constraints with v_2. v_3·v_2 = a(v_1·v_2) + b(v_2·v_2) + w·v_2 = -2a + 2b = -5. (Equation 2) Step 4: Solve for a and b. From Equation 1: b = 2a - 3. Substituting into Equation 2: -2a + 2(2a - 3) = -5 ⟹ -2a + 4a - 6 = -5 ⟹ 2a = 1 ⟹ a = 1/2. Then b = 2(1/2) - 3 = -2. Step 5: Find the orthogonal component. v_3 = (1/2)v_1 - 2v_2 + w. Then v_3·v_3 = (1/4)(4) + 4(2) + 2(1/2)(-2) - 2(1/2)v_1·v_2 - 2(-2)v_2·v_1 + w·w = 1 + 8 - 2 + 2 + w·w = 9 + w·w = 29. Thus w·w = 20. Step 6: Verify options. For option (b): v_3 = 3i - 2j ± 4k. Check v_3·v_3 = 9 + 4 + 16 = 29 ✓. Check v_3·v_1: requires knowing v_1 explicitly, which is not given. Without explicit vector representations, we cannot fully verify all conditions. The problem statement is incomplete as v_1, v_2 are not explicitly defined in terms of components. ∴ Answer: Unknown
Correct Answer: Unknown